Single correct option
The sum of an infinite geometric progression is 18 and the sum of the squares of its terms is 108. What is the first term?
- A. 6
- B. 8
- C. 9
- D. 12
Show answer and solution
Answer: (C) 9
a/(1 − r) = 18, and the squares form a GP with first term a² and ratio r², so a²/(1 − r²) = 108.
Dividing the second by the first: a/(1 + r) = 108/18 = 6.
So a = 18(1 − r) = 6(1 + r) ⇒ 18 − 18r = 6 + 6r ⇒ r = 1/2, and a = 9.
Check: 9/(1/2) = 18 ✓; 81/(3/4) = 108 ✓.
Dividing the second by the first: a/(1 + r) = 108/18 = 6.
So a = 18(1 − r) = 6(1 + r) ⇒ 18 − 18r = 6 + 6r ⇒ r = 1/2, and a = 9.
Check: 9/(1/2) = 18 ✓; 81/(3/4) = 108 ✓.
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